Hardy’s inequalities for monotone functions on partly ordered measure spaces

The theory of weighted inequalities for the Hardy operator, acting on monotone functions in R+, was first introduced in [2]. Extensions of these results to higher dimensions have been considered only in very specific cases: in particular, in the diagonal case, for p = 1 only (see [3]). The main difficulty in this context is that the level sets of the monotone functions are not totally ordered, contrary to the one-dimensional case, where one considers intervals of the form (0, a), a > 0. It is also worth pointing out that, with no monotonicity restriction, the boundedness of the Hardy operator is known only in dimension n = 2 (see [11, 13] and for an extension in the case of product weights see [4]). In this work we characterize completely the weighted Hardy inequalities for all values of p > 0, namely, the boundedness of the operator